Keith Promislow

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We use renormalization group (RG) techniques to prove the nonlinear asymptotic stability for the semi-strong regime of two-pulse interactions in a regularized Gierer-Meinhardt system. In the semi-strong limit the localized activator pulses interact strongly through the slowly varying inhibitor. The interaction is not tail-tail as in the weak interaction(More)
Using a standing light wave potential, a stable quasi-one-dimensional attractive dilute-gas Bose-Einstein condensate can be realized. In a mean-field approximation, this phenomenon is modeled by the cubic nonlinear Schrödinger equation with attractive nonlinearity and an elliptic function potential of which a standing light wave is a special case. New(More)
We employ global quasi-steady manifolds to rigorously reduce innnite dimensional dynamical systems to nite dimensional ows. The manifolds we construct are not invariant, but through a renormalization group method we capture the long-time evolution of the full system as a ow on the manifold up to a small residual. For the parametric nonlinear Schrr odinger(More)
We show the soliton solutions of the integrable Manakov equations exhibit an instability under arbitrarily small Hamiltonian perturbations. The instability arises from eigenvalues embedded in the essential spectrum of the associated linearized operators; these eigenvalues are dislodged by smooth perturbations. Speciically we consider perturbations which(More)
We demonstrate the existence of localized oscillatory breathers for quasi-one-dimensional Bose-Einstein condensates confined in periodic potentials. The breathing behavior corresponds to position oscillations of individual condensates about the minima of the potential lattice. We deduce the structural stability of the localized oscillations from the(More)
We consider the linear stability and structural stability of non-ground state traveling waves of a pair of coupled nonlinear Schrr odinger equations (CNLS) which describe the evolution of co-propagating polarized pulses in the presence of birefringence. Viewing the CNLS equations as a Hamiltonian perturbation of the Manakov equations, we nd parameter(More)